Question

Prove that \( 2+3 \sqrt{5} \) is an irrational number.

Answer 1

Let us assume that 2 + 3√5 is rational .Then, there exist positive co primes a and b such that

\(2 + 3\sqrt5 = \frac{a}{b}\)

\(3\sqrt5 = \frac{a}{b} - 2\)

\(\sqrt5 = \frac{\frac{a}{b} - 2}{3}\)

\(\sqrt5 = \frac{a-2b}{3b}\)

This contradicts the fact √5 is an irrational number 

Hence 2 + 3√5 is irrational.